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By Ian W. Knowles, Roger T. Lewis

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Baras and J. A . Goldstein, In preparation. [2] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, Corn. Math. Phys. 80 (1981), 59-98. [3] C. L. Fefferman, The Uncertainty Principle, Lecture Notes for the h e r . Math. SOC. Colloquium Lectures, Denver, Colorado, January, 1983. [4] H. -W. Schmincke, J. Walter, and R. W%t, On the spectral theory of Schradinger and Dirac operators with stronEly singular potentials, in Spectral-Theory and Differential Equations-(ed.

Can be replaced by (Let us add parenthetically that the domain C:(RN\{O)) 2 N 2 5. ) Recently Baras and Goldstein [I] proved a result which, among other things, XI-^ C i ( R N ) if provides an alternate proof and strong sharpening of the second assertion of Theorem l(i). Of concern is the heat (rather than the Schrodinger) equation au/at + u + f(x,t) 1x1 T > 0 , c > 0 , and 0 = AU + (1) 1 where S f E L (RN X ( 0 , T ) ) are given. A solution in the sense of distributions is sought whose initial value is a (nonnegative) finite Radon measure LI in the sense that for (x,t) for each E RN x ( 0 , T ) @ E C;(W N ) .

N. Harrington - THEOREM [9]. which {-I. L e t F ( z ) be a r a t i o n a l f u n c t i o n , o f degree g r e a t e r t h a n one, f o r belongs t o an a t t r a c t i v e o r i n d i f f e r e n t k - c y c l e and f o r which J c iR u The a s s o c i a t e d J a c o b i m a t r i x d i s s e l f - a d j o i n t w i t h spectrum J and s p e c t r a l d e n s i t y p, t h e balanced measure f o r F(z). EXAMPLE. L e t F ( z ) = z 2 -A with X > 2. The c o n d i t i o n s o f t h e l a t t e r theorem a r e met, J b e i n g a Cantor s e t i n R .

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